The Supreme Court upheld a key part of Arizona’s tough anti-illegal immigration law in a 5-3 decision on Monday that allows police officers to ask about immigration status during stops. That part of the law, which never went into effect because of court challenges, will now immediately be enforced in Arizona. Other parts of the law, including a provision that made it a state crime for illegal immigrants to seek work, will remain blocked, as the justices affirmed the federal government’s supremacy over immigration policy.

Justice Anthony Kennedy, the court’s swing vote, wrote the opinion, and was joined by Chief Justice John Roberts, Ruth Bader Ginsburg, Stephen Breyer and Sonia Sotomayor. Conservative Justices Antonin Scalia, Samuel Alito and Clarence Thomas partially dissented, saying the entire law or most of the law should have been upheld.

In the opinion, Justice Kennedy wrote that the federal government’s “power to determine immigration policy is well settled.” But he also showed concern for what he described as Arizona’s outsize burden in dealing with illegal immigration, seeming to sympathize with the state’s decision to butt in on immigration enforcement. “Arizona bears many of the consequences of unlawful im­migration,” he wrote. “Hundreds of thousands of deportable aliens are apprehended in Arizona each year.” But, ultimately, the justices found that Arizona cannot mete out its own state punishments for federal immigration crimes.

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“Arizona may have under­standable frustrations with the problems caused by illegal immigration while that process continues, but the State may not pursue policies that undermine federal law,” Kennedy wrote in the opinion’s conclusion.

The police immigration checks are allowed, however, because state police would simply flag federal authorities if they found an illegal immigrant. Kennedy did not rule out that these checks may be implemented in an illegal way, which means more lawsuits may be forthcoming.